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Chapter 6: Problem 89

Write as a percent. Write the remainder in fractional form. $$\frac{2}{9}$$

Short Answer

Expert verified
The fractional form of 2/9 expressed as a percentage is 22.22%, and the remainder in fractional form is \(11/50\).

Step by step solution

01

Convert Fraction to Decimal

To convert a fraction into a percentage, the fraction must first be converted into a decimal. This is achieved by dividing the numerator (top number) by the denominator (bottom number). For this exercise, \(2 \div 9 =0.2222... \).
02

Conversion to Percent

To convert this decimal to a percentage, the decimal must be multiplied by 100. In this case, \(0.2222... \times 100 = 22.22\) percent.
03

Finding the Remainder

To find the remainder of the fractional form, subtract the percentage (22) from the whole (100) and express this as a fraction over the whole (100). In this case, \(22.22 - 100 = -77.78\), but we do not have negative remainder. So integer part 22 is 100% correct rest 0.22 part or 22% we need to find as fraction. From decimal part 0.22 we can get 22/100 so simplified gives us \(11/50\). This is the remainder in fractional form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fraction to Decimal Conversion
When asked to convert a fraction to a decimal, the process primarily involves division. You take the numerator (the top part of the fraction) and divide it by the denominator (the bottom part of the fraction). In this way, the operation of dividing 2 by 9 gives us the decimal 0.2222... which is a repeating decimal. This means the number 2 repeats infinitely after the decimal point.
  • First, rewrite the fraction as a division problem: \(\frac{2}{9} = 2 \div 9\).
  • Perform the division to arrive at 0.2222..., or approximately 0.22 when rounded to two decimal places.
Whenever you deal with repeating decimals, noting the pattern helps in further calculations.
Percentage Calculation
Once you've converted a fraction into a decimal, the subsequent step towards expressing it as a percentage involves a simple multiplication. You multiply the decimal by 100 to transition it into a percentage.
  • Take the decimal representation, like 0.2222..., and multiply by 100: \(0.2222... \times 100 = 22.22\).
  • This gives you 22.22%, which signifies the percentage equivalent of the original fraction \(\frac{2}{9}\).
This calculation often involves rounding the decimal to a manageable number of decimal places. Remembering that multiplying by 100 shifts the decimal point two places to the right, can make this process intuitive.
Remainder as a Fraction
In some contexts, especially when dealing with percentages, it becomes necessary to represent remainders or fractional parts. While the percentage provides a whole number part, sometimes parts of it need to be broken down further.
  • From our calculation, 0.22 as the decimal part of 22.22% corresponds to the leftover percentage.
  • Express this fractional component by considering it as \(\frac{22}{100}\), a fraction representing 22% of a whole.
  • To simplify, divide both the numerator and the denominator by their greatest common divisor, resulting in \(\frac{11}{50}\).
This remainder representation is an elegant way to express parts of a percentage that don't form a whole number, offering precision in mathematical communication.

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